From torpid to rapid mixing: group averaging for a weakly interacting Ising star

We study lazy single-site Metropolis dynamics $P_β$ at inverse temperature $β\\geq 0$ on $\\{-1,+1\\}^d$ for an Ising star with additional signed interactions among the leaves. If the absolute row sums of the leaf-interaction matrix are at most $κ\\le1/2$, the worst-case total-variation mixing time of $P_β$ is at least of order $d\\exp\\{cβ(d-1)\\}$ for $β\\ge1$, with universal $c>0$. Averaging over global spin reversal reduces the mixing time to $\\mathcal O(d^2(1+β))$ for both $GP_βG$ and $(P_β+G)/2$, where $G$ is the Gibbs kernel induced by the partition of the state space into spin-reversal orbits $\\{-x,x\\}$. Partition-function interpolation gives the lower bound for $P_β$. The upper bounds follow from Wu's Dobrushin inequality and a decomposition into projection and restriction chains. This gives an explicit example in which group averaging turns an exponentially slow chain into a polynomially fast one.